Problem 36 - College Algebra

Question

Use the formula for the sum of the first \(n\) terms of an arithmetic series to find the sum. $$ -1+3+7+\ldots+31 $$

Step-by-Step Solution

Answer

The sum is 135.

Step 1: Identify the variables

In an arithmetic series, the sequence starts with a first term \(a\) and has a common difference \(d\). From the series, identify the first term \( a = -1 \). The second term is 3, so the common difference \( d = 3 - (-1) = 4 \). The last term given is 31.

Step 2: Find the number of terms (n)

The nth term of an arithmetic sequence can be found with the formula \( a_n = a + (n-1)d \). Set \( a_n = 31 \) and solve for \( n \): \[ 31 = -1 + (n-1)\times 4 \] Simplify to find \( n \): \[ 31 + 1 = 4n - 4 \] \\[ 32 + 4 = 4n \] \\[ 36 = 4n \] \\[ n = 9 \].

Key Concepts

First Term (a)

The first term of an arithmetic series serves as the starting point of the sequence. It is often denoted by the letter \(a\). Identifying this term is crucial, as it helps set up the entire series. In our example, the first term \(a\) is \(-1\). This value is the initial number in the sequence \(-1, 3, 7, \ldots, 31\). Recognizing the first term allows us to understand the base from which all other terms are calculated. The first term is a cornerstone in arithmetic sequences, especially when combined with the common difference \(d\) to identify and calculate further terms.

Common Difference (d)

In an arithmetic series, the common difference \(d\) is the consistent amount between any two consecutive terms. This means if you subtract the first term from the second term, the result is the common difference. It is necessary for understanding the pattern of change from one term to the next.
For instance, in the sequence \(-1, 3, 7, \ldots \), the common difference \(d\) can be found by calculating \(3 - (-1) = 4\). Every subsequent term is derived by adding this common difference of 4 to the previous term.

• Second term: \(-1 + 4 = 3\)
• Third term: \(3 + 4 = 7\)

Knowing \(d\) is essential as it simplifies the process of finding other terms in the series.

Sum of the First n Terms

The sum of the first \(n\) terms of an arithmetic series is a handy calculation and uses a particular formula: \[S_n = \frac{n}{2} (2a + (n-1)d)\]This formula allows you to calculate the total of a given number of terms in a sequence without having to manually add them all up.
First, find \(n\), the number of terms. You can use the formula for the \(n\)-th term in an arithmetic series: \[a_n = a + (n-1)d\] Setting the last term \(a_n = 31\), and solving for \(n\), we find \(n = 9\).
Finally, plug the values \(a = -1\), \(d = 4\), and \(n = 9\) into the sum formula:

• Calculate the expression inside the parentheses \(2(-1) + (9-1) \times 4 = -2 + 32 = 30\)
• Now, \(S_9 = \frac{9}{2} \times 30 = 135\)

Thus the sum of the first 9 terms of this arithmetic series is 135. This methodical approach ensures accuracy and efficiency in finding the sum.